The matching with contracts model (Hatfield and Milgrom 2005) is widely considered to be one of the most important advances of the last two decades in matching theory. One of their main messages is that the set of stable allocations is non-empty under a substitutes condition. We show that an additional irrelevance of rejected contracts (IRC) condition is implicitly assumed throughout their analysis, and in the absence of IRC several of their results, including the guaranteed existence of a stable allocation, fail to hold. (JEL C78, D86)

In 2012 Brazilian public universities were mandated to use affirmative action policies for candidates from racial and income minorities. We show that the policy makes the students' affirmative action status a strategic choice, and may reject high-achieving minority students while admitting low-achieving majority students. Empirical data shows evidence consistent with this type of unfairness in more than 49% of the programs. We propose a selection criterion and an incentive-compatible mechanism that, for a wider range of similar problems and the one in Brazil in particular, removes any gain from strategizing over the privileges claimed and is fair.

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